Imagine a system that runs as normally (either first past or a sensible voting system) unless >50% of eligible voters fail to vote or vote 'none' or something like that, and then a random citizen is put in power.
This is actually interesting, but I'd change it a bit. Every voter who doesn't vote is treated as a vote towards "none", and candidates must defeat the "none" category as well. If "none" wins, then that means none of the candidates won, and you resolve it through some other means (like lottery).